COMPLEX NUMBERS AND DE MOIVRE'S THEOREM SYNOPSIS. Ay umber of the form x+iy where x, y R ad i = - is called a complex umber.. I the complex umber x+iy, x is called the real part ad y is called the imagiary part of the complex umber. 3. A complex umber is said to be purely imagiary if its real part is zero ad is said to be purely real if its imagiary part is zero. 4. (a) Two complex umbers are said to be equal if their real parts are equal ad their imagiary parts are equal. (b) I the set of complex umbers, there is o meaig to the phrases oe complex is greater tha or less tha aother i.e. If two complex umbers are ot equal, we say they are uequal. (c) a+ ib> c + id is meaigful oly whe b = 0, d = 0. 5. Two complex umbers are cojugate if their sum ad product are both real. They are of the form a+ib, a-ib. 6. cisθ cisθ = cis(θ + θ ), cis θ cis θ 7. 8. a + ib a + ib = + i = i i, i + i = i ( a a b ) + ia ( b a b ) a = cis(θ θ ), = cosθ isiθ. cosθ + isiθ 9. x + y is called the modulus of the complex umber x + iy ad is deoted by r or x + iy. 0. Ay value of θ obtaied from the equatios cosθ = x r, siθ = y r is called a amplitude of the complex umber.

.i) The amplitude lyig betwee -π ad π is called the pricipal amplitude of the complex umber. Rule for choosig the pricipal amplitude. ii) If θ is the pricipal amplitude, the -π<θ π. If α is the priciple amplitude of a complex umber, geeral amplitude = π + α where Z. 3.. Z + Z Z + Z,. Z - Z Z + Z 3. Z + Z Z Z 4. Z - Z Z - Z 5. Z Z = Z Z 6. Z Z Z = Z 7. Z = Z 8. Z + Z = Z + Z 9. Z Z = Z Z 0.Z Z = Z. Z. Z Z = Z 4. The least value of Z-a + Z-b is a-b. 5. si(ix) = i si hx cos ix = coshx ta ix = i tahx cot ix = - i cothx (i), e -ix = cosx + i six isix = e ix ix e = sih(ix) 7 cosx = e ix e 6. i) Amp (Z Z ) = Amp Z + AmpZ ii) Amp Z Z = AmpZ AmpZ ix + =cosh(ix) π-θ θ -(π-θ) -θ iii) Ampz + Amp z = π (whe z is a egative real umber) = 0 (otherwise) 7. * r(cosθ + i siθ) is the modulus amplitude form of x+iy. * = cos0 + i si 0 * = cosπ + i si π

* i = cos π + i si π iπ * If z = r (cosθ + i siθ), the, log Z = log r + iθ ad log i =. 8. If the amplitude of a complex umber is π, its real part is zero. 9. If the amplitude of a complex umber is π, its real part is equal to its imagiary 4 part. 0. Every complex umber x + iy is regarded as a poit (x, y) i the Argad diagram.. If P represets a complex umber i the Argad diagram, the OP represets modulus of the complex umber ad XOP represets the amplitude of the complex umber.. If Z = k the locus of the poit represetig Z i the Argad diagram is a circle coditio Z - Z = k circle Z Z Z Z Z Z Z Z = k ( ) circle locus of the poit represetig Z = lie i.e., the perpedicular bisector of the lie joiig Amp Z = α(a costat) Amp (Z-Z ) = α Z & Z. lie lie π 3π 3. i) If Z + Z = Z - Z, the amplitudes of Z, Z differ by a right agle or. 4. ii) If Z + Z = Z + Z, the amplitudes of Z, Z differ by 0 orπ. + cosθ+ isi θ = cosθ + i siθ + cosθ isi θ

5. + si θ+ icosθ = cos π π θ θ + si θ icosθ + isi. 6. If is ay iteger, (cosθ + i siθ) = cos θ + i si θ 7. If is ay fractio, oe of the values of (cosθ + i siθ) is cos θ + i si θ. 8. (siθ + i cosθ) π π = cos θ isi θ 9. 30. + cosθ + isiθ cosθ isiθ + + siθ + icosθ siθ icosθ + = cosθ+ i siθ π π = cos θ + isi θ 3. If x = cosθ + i siθ, the x + = cosθ, x = i siθ x x x + = cos θ, x x = i si θ. x 3. If cosa + cosb + cosc = 0, sia + sib + sic = 0 the cos3a + cos3b + cos3c = 3 cos(a+b+c) si3a + si3b + si3c = 3 si(a+b+c) cos A + cos B + cos C = 3 si A + si B + si C = 3 A - B = π π π, B - C =, C - A = 3 3 3 { } 33. If Z = r cos( kπ + θ) + isi( kπ + θ) (θ beig the pricipal amplitude) the values ( ) of Z are give by r cis ( kπ + θ) / where k = 0,,,... -. 34. The th roots of a complex umber form a G.P. with commo ratio cis π which is deoted by ω.

35. The poits represetig th roots of a complex umber i the Argad diagram are cocyclic. 36. The poits represetig th roots of a complex umber i the Argad diagram form a regular polygo of sides. 37. The poits represetig the cube roots of a complex umber i the Argad diagram form a equilateral triagle. 38. The poits represetig the fourth roots of complex umber i the Argad diagram form a square. 39. The th roots of uity are, w, w,... w - where w = cis π 40. The sum of the th roots of uity is zero (or) the sum of the th roots of ay complex umber is zero. 4. The cube roots of uity are,ω,ω where ω = cis π, ω = cis 4 π or 3 3 ω = 4. + ω+ ω = 0. 43. ω 3 = + i 3 i, ω = 3. 44. The product of the th roots of uity is (-) -. 45. The product of the th roots of a complex umber Z is Z(-) -. 46. ω, ω are the roots of the equatio x + x + = 0 / / / 47. The cube roots of ay real umber a are a a w a w 3 3 3,,. 48. ABCD is a square i the Argad diagram ad the cetre of the square is Z. If A represets Z, the poit B represets the complex umber Z + i(z -Z) i the Argad diagram. 49. If Z, Z, Z 3 are the complex umbers represeted by vertices of a equilateral triagle i the Argad diagram the Z Z Z Z Z Z Z Z Z + + =. 3 3 3 0

50. Z,Z ad the origi are the vertices of a equilateral triagle. The Z + Z 5. If q is ot a multiple of, the sum of q th powers of th roots of uity is zero. If q is a multiple of the sum is. 5. (a w) (a w ) (a w 4 )(a w 8 ) = (a + a + ) 53. If, w, w...w - are the th roots of uity the (-w)(-w )(-w 3 )...(-w - ) =. 54. If z, z, z 3 are the vertices of a equilateral triagle the the followig coditios are true. a) z + z + z 3 = z z _ z z 3 + z 3 z b) (z z ) + (z - z 3 ) + (z 3 z ) = 0 + + c) z z z z3 z3 z = 0 d) z + wz + w z 3 = 0 55. Logarithm of a complex umber ) log (a + ib) = ½ log(a ) + I ta - (b/a) ) log (a ib) = ½ log (a ) I ta - b/a z 3) log z = log + I amp z 4) log ( + i)= ½ log + I π / 4 5) log i = i π / 6) log i = / e π 56. Relatios betwee trigoometric ad hyperbolic fuctios ) cos h (ix) = cos x ) si h (ix) = i si x 3) ta h (ix) = i ta x 4) cot h (ix) = -i cot x 5) sec h (ix) = sec x 6) cosec h (ix) = -i cosec x Z Z =

7) cos ix = coshx 8) si ix = coshx 9) ra ix = i ta hx 0) cot ix = i cot hx ) sec ix = sec hx ) cosec ix = - i cosec hx 57. If z is a complex umber lyig o a circle with radius r the the complex umber 58. az+b lies o a circle with radius ar. a + ib = ± a ib = ± a a + a + i + a i 59. z + z = z + z + zz + zz a a 60. z + z + z z = ( z + z ) a a 6. ω, ω are complex cube roots of uity, the ω +ω = if is a +ve iteger which is a multiple of 3 = if is a +ve iteger which is ot a multiple of 3 6. a) If z is o real, Amp z = θ, Ampz = - θ b) Amp ( ) z z 63. a) If amp z z z z = Amp z - Ampz = ±, the the locus of z is the circle o the lie joiig π z,z as diameter. z z b) If amp = 0or π, the the locus of z is a straight lie passig z z through z,z.

64. The equatio of the perpedicular bisector of the lie segmet joiig the poits which represet z ad zz z + z z z = z z z is ( ) ( ) 65. Area of the triagle o the argad plae formed by the complex umbers z, iz ad z+izis z. 66. If z, z, z3 are represeted by poits A, B, C i argad plae, the BAC = arg z z 3 z z 67. The legth of the perpedicular from a poit z to the lie a z + az = 0 is give by a z + az a 68. If a = cosα + i si α, b = cosβ + i siβ ad c = cosγ + i siγ ad a+b+c = 0, the + + = 0 a b c 69. The poits represetig the roots of the equatio ( z ) = z where is a positive iteger i the argad diagram lie o a lie whose equatio is x = ½. 70. If z is ay o zero complex umber, the the area of the triagle formed by the complex umbers z, ωz ad z+ωz as its sides is 3 z 4 7. The complex umbers z,z ad the origi form a isosceles triagle with vertical agle α if z z = e iα 7. If α is a o real th root of uity, the +α+3α +..+. α - = α 73. If the complex umbers z, z, z 3 represet the vertices of a equilateral triagle such that z = z = z 3, the z + z + z 3 = 0 74. The equatio b z z = c where b is a o-zero complex costat ad c is real, represets astraight lie.